# A Critique Of Pseudo-Cardinal Utility Functions

Okay, I’ve put this one off for long enough…

A while back, I explained why social welfare functions don’t exist. In this post, I also described how neoclassical economics has abandoned the idea of cardinal welfare functions in favor of ordinal utility functions due to the fact that cardinal utility functions cannot withstand legitimate criticisms.

As a quick reminder, a utility function is a numerical mapping of preferences. I would apply a value of 100 for a good milkshake and 5 for a bad milkshake, for example. An ordinal utility function has it that the good milkshake is preferred to the bad milkshake, but this is all we can derive from the numbers provided; we cannot tell how much I would prefer a good milkshake to a bad one. On the other hand, a cardinal utility function actually allows us to form ratios and calculate this: I would gain 100 units of utility from a good milkshake, but only 5 from a bad milkshake.

Now, the problem here is that there is no measure of utility as such, so we cannot create cardinal utilities, leaving us with only ordinal utilities.

But that is not the end of the story. Instead of using some made-up ‘util’ measure, some economists attempt to create a cardinal utility function by using money as a proxy for utility. You see this often in mixed-strategy game theory, as a matter of fact, but also in other contexts. This is what I would call a pseudo-cardinal utility function: you assume that utilities as such are still ordinal, but you use a cardinal proxy to determine the changes in utility. An easy way to think of it is like this: you currently have a utility of x. If I give you a \$5 transfer, you will have a utility of x + y, where y is the utility gained from an additional \$5. If I give you a \$10 transfer, you will have a utility of x + 2y.

But if you look closely, there’s an inherent problem here:  money is subject to diminishing returns.  Imagine it this way:  if you tax a rich man to the tune of \$1000, it bothers him less than if you tax a man whose net value is \$1000 by the same amount.  Thus, the utility derived from an additional \$5 is actually dependent upon the current utility, and you need to measure the utility in a consistent manner over values.  In other words, you’re back to a cardinal utility function.

So basically, pseudo-cardinal utility functions either avoid the fact that there are diminishing returns to money or they implicitly assume a cardinal utility function to piggyback off of.  In either case, they are not acceptable for serious use, any more than calculating the number of utils is acceptable.

## 13 thoughts on “A Critique Of Pseudo-Cardinal Utility Functions”

1. Michael A. Lewis says:

It seems to me that diminishing returns to money is a plausible assumption to make. But is there any evidence to support it? I suppose one would have to measure, in some way, the degree to which taxes “bother” someone, in order to determine if they “bother him less”). To say that x bothers person 1 less than it does person 2 is certainly a mathematical statement about differences in the degree of something. The question is what is that “something?” Is it a cardinal “something,” ordinal “something,” or none of the above? It seems that you may be questioning pseudo-cardinal utility functions on the basis of an assumption, which may rely on the notion of cardinal utility to make it plausible. I say “may” because it isn’t clear what it is you have in mind, which one could measure to arrive at the conclusion of diminishing returns to money.

2. To show that money does follow the law of diminishing marginal returns, I ask you to consider this: pretty much all people work, generally 8 hours a day or thereabout. Nobody works 24 hours a day, and in a relatively wealthy society, you see people working many fewer hours than they theoretically could and historically have. But at the same time, we still work for a living. This implies that money is more valuable than leisure to some extent (because slacking won’t pay the bills), but that there are diminishing marginal returns because we eventually decide to kick up our feet and enjoy some leisure.

As an anecdotal example (which in itself proves nothing, but I think does give a fair insight into human behavior by looking at this human’s behavior), I had the choice between two jobs, one paying about \$4000 a year more than the other. I turned down the better-paying job in part because I was almost guaranteed to work at least three or four hours of (unpaid) overtime a week and some 55+ hour work weeks. Even though the average wage still turned out to be higher at the higher-paying job, I valued 3 to 4 hours a week as worth more than the \$4000 I turned down. On the other hand, if I had a third job opportunity in which I could trade in another \$4000 for 3 to 4 more hours a week off, I would not do so, simply because I would prefer the extra money. Now, other people may have taken the better-paying job, and there are some people who basically live for work, folks like investment bankers. But even they don’t work all the time.

In the above example, I don’t use a cardinal utility function—or any type of utility function, really—to make a point. You could rephrase my statement in terms of an ordinal utility function as a pedagogical exercise, but that’s entirely different from relying upon one to make a point. Nor would you, the external observer, need to measure anything to see that I, the acting entity, follow the aforementioned law of diminishing marginal returns, at least when it comes to money earned via work. All you have to do is observe that I go in, work 40 hours a week, and go home to do non-work-related things which do not earn me money.

The point of this post was mostly a dig at the airs to precision in mixed-strategy game theory and many macoreconomic models by using utility as though it were a cardinal value while simultaneously attempting to claim that utility—rightly—cannot be a cardinal value. Sorry for the lack of clarity on that one; hopefully this gives you a better idea of what I’m trying to argue, and thanks for your thoughts.

3. Michael A. Lewis says:

4. You would need to count the number of hours—fair enough. You wouldn’t need to calculate some concept of utility, though.

My reason to accept the \$4K for hours 36-40 are because the opportunity cost is outweighed by the value of the goods I can purchase by continuing to work. For a quick and somewhat silly example, let’s plot a rough estimate of my opportunity costs. Imagine that I only receive income from work (which isn’t entirely true, but it simplifies things). Meanwhile, my leisure alternative is, say, sitting around and playing video games. Here is how I would arrange things, based in terms of satisfying my most-pressing needs first:
Hours 3-12: Paying rent.
Hour 13: Auto insurance.
Hours 14-19: Student loans.
Hours 20-22: Utilities.
Hours 23-26: Various subscriptions (internet, telephone, etc.).
Hours 27-30: Various unexpected purchases and long-term averaging of expected purchases (clothing, automobile and repairs, household appliances).
Hours 31-38: Savings, rainy day fund, etc.
Hours 39-40: Video game.
Hours 41-43: Other forms of entertainment (more video games, sporting events, etc.).
Hour 44: Purchasing bootleg copies of Cheers.

As a side note, this follows extremely closely with Carl Menger’s description of how a theoretical farmer allocates bags of corn on his farm, starting with the most-vital and pressing concerns and moving out to those which he judges to be less important. Thus, he ensures his own livelihood before making sure that he has enough of a crop for the next year, and only after that does he start taking care of livestock for his personal gain (eggs, milk, meat, etc.). If you’d like a more detailed explanation of that, go to http://www.mises.org/about/3239 and search for “sack of grain” about halfway down the page.

Throughout this, you can see that the most pressing need in this example is food, followed by shelter and transportation. That is what I would spend the money on. If I decide only to work 2 hours a week, I would have to play video games in a cardboard box, and because I would prefer not to do that, I am willing to forego that much leisure in order to live in a slightly larger and less corrugated place. Eventually, however, I’m down to buying bootleg copies of Cheers, a series I didn’t even like, and at that point, I’d rather just sit at home and get the snot beat out of me online by some 14-year-old kid who’s memorized the entire game. Thus, working up until hour 44 is less worthwhile for me than the opportunity cost (staying at home and playing video games, as in our silly and simple example). But at hour 36, I would still prefer to save money, so I would still like to work a little bit longer. That is why I would turn down offer 1 (44 hours, gaining \$4000 more a year) and offer 3 (36 hours, losing \$4000 a year) in favor of offer 2 (40 hours), and why money has diminishing returns: the stuff that you can get with money becomes relatively less valuable compared to your leisure-filled alternatives.

And because opportunity cost is an entirely subjective phenomenon, you can see people like corporate lawyers (your example) and investment bankers (my example) whose opportunity costs are relatively less worthwhile, and thus are willing to work like dogs. This has nothing inherently to do with the labor supply, but is instead dominated by personal preferences. [You may also point out that some hard-working people gain additional enjoyment by virtue of owning their own company, or that extremely hard work now will set you up for a nicer career and more leisure later on, but they still fit in to the same idea.]

In my example above, you see a bachelor who doesn’t care that much about purchasing fancy things. An investment banker, meanwhile, will probably have more expensive desires—nice cars, nice clothing, jewelery, that kind of thing—and their desire to purchase such products necessitates working longer hours. If our intrepid hero above were to get married and start a family, his priorities change and maybe he would be willing to work additional hours because taking care of the children—an expensive act—becomes more important but he still wants to save money. Or maybe his wife is one of those corporate lawyers and he decides to cut back on the number of hours in order to take care of the children. Or maybe they both just keep working the same number of hours as before because things work out best that way. This depends solely upon the value judgments of the individual actors given their perceptions of the opportunities available and what they would forego given each alternative.

With the example on taxation, I purposefully chose a ridiculous example, one which made my point without any chance of potentially being wrong and also served as an extreme regressive tax. In the example, I noted that taking \$1000 away from a person who makes a lot is of less harm than taking \$1000 away from a man who only has \$1000 (i.e., taking everything away from him). In the hourly breakdown example above, supposing that I work 40 hours a week, imagine if a new tax effectively took away 2 hours of my time without providing any subsequent benefit for me. At 40 hours, this means I no longer will purchase a video game that week. This would leave me worse off than before, but it would not be nearly as bad as if I were only working 2 hours a week and couldn’t shift into doing additional labor. Alternatively, if that’s a little too out-there, imagine that I’m working the 40 hours a week with this breakdown, but the tax is so high that I need all 40 hours just to pay the taxman or that I make so little money that it takes all 40 hours just to pay for food. At that point, I would no longer be able to afford food. In the first case, I have to make do with older video games, but in the second case, it’s the end of me because I starve. I’m still the same person with the same set of needs/desires, but where I stand financially will have an effect on how much a lump sum tax with no benefit coming from it hurts me because it will take away different goods.

Now that was a mouthful… But I’ll try to sum up my argument in the original post again, just to be extra-clear about things. I personally am not a big fan of utility theory—it’s in my Austrian roots not to like it very much. I have grown to accept truly ordinal utilities as a teaching exercise to try to teach students some of the finer points of why people behave as they do, but I think that ordinal utilities lead people to try to pin down precisely the effects of particular policies or actions with macoreconomics, or to predict behavior in the case of game theory. When economists start doing this, even the well-trained ones can sometimes forget that the numbers which they chose really don’t have a cardinal value. This remains true when you use money (which is the general game theory copout) because, as I noted in the example above, the money that I earn in 2 hours of work actually will have different values for me depending on how many hours I have already worked. If we’re talking the first two hours, the money has an extremely high value; if we’re talking hours 39 and 40, the value is relatively less. So if you have a game theoretic example in which I earn a certain payout in terms of money, the actual value (to me) of the money I earn by playing the game will not match 1 for 1 changes in the payoff, but will instead depend upon the marginal gain of that payoff, i.e., the subjective value of whatever it is that I purchase.

I hope that clarifies things a little bit more, and I am interested in hearing your thoughts if the above makes any sense.

5. Michael A. Lewis says:

6. Leisure as a good is a common notion in economics, just like apples, oranges, and so on. It’s kind of a catch-all term for “stuff you do but don’t get paid for.” And if you’re fine with the notion of demand functions in general, a demand function for leisure would be no different. There are certain groups of economists (again, mostly the Austrians) who don’t like the idea of calling them functions because that implies continuity and knowledge of the entire range of hypothetical relative price situations. From what I’ve gathered, you’re not in that group, so go right ahead and think of it as a leisure function.

As an economist, talking too much about “needs” is self-defeating. The idea of a “need” implies that, well, everybody needs a thing. Outside of a few extremely simple examples (which is what I tried to get at with the “most-vital and pressing concerns” followed by “those judged less important”), we should be talking about demand, a term which is more subjective and fits better with economics. Doing this tends to avoid a lot of pop sociology/psychology fallacies which are plenty tempting for a lot of people (including economists). Basically, economics cannot really give you a good answer to why people demand particular things. Gary Becker tried this in the 1970s, but succeeded in just pushing the question back one step.

Because of this, I would not try to phrase the brick-layer versus investment banker quandry in the form of “needs,” but rather “demand.” What you bring up isn’t a contradiction, but rather a statement that the demand and opportunity cost of brick-layers differ from investment bankers. After 40 hours, our hypothetical brick-layer is at a point where the value earned from working another hour is less than the value of leisure, whereas for the investment banker, the value from that 41st hour of work is still greater than the value of that hour of leisure (assuming, as we are still in a simple analytical world, that each of them is legally allowed to work as many hours as that person wants and that the person can find employment to work that many hours. Real life is more complex, but the principle is what we’re discussing).

Increasing returns to earnings would imply that people will purchase less valuable goods before they buy more valuable things, given that both are available and affordable. This would not make sense for a rational person, especially once we throw in that (from an economist’s perspective) valuation is solely a subjective criterion, something which the actor alone chooses. It would be like, in my example above, me purchasing video games before I’ve paid the rent. I value housing more than games*, so there’s no reason that I should do so. But in order to have increasing marginal returns, something like this would need to happen. So I would politely suggest not getting too caught up in trying to determine what, specifically, constitutes a “need,” the relative value of a “need” versus a “want,” and so on. Economic theory really doesn’t handle this side of things, but instead assumes that people, for whatever reason, have some more or less defined set of desires as well as a notion of how these desire rank in comparison to one another, and from this we get demand. It’s kind of nebulous when you get that far out, but most economists punt on this, saying that it’s really a psychology issue. As noted above, Gary Becker did take a stab at it. Viktor Vanberg summarized and subsequently criticized his argument rather effectively in Rules and Choice in Economics (chapter 3), so if you can get your hands on a copy, I would highly recommend it, and not just because I wrote my thesis under him…

* – Well, okay, I simplified the example above by not taking into account marginal differences, but if I were to do that, then we’re talking about a significantly more complex example which provides no real additional insight.

7. Michael A. Lewis says:

8. Michael A. lewis says:

I said increasing, “marginal utility of income.” I meant to use your language of increasing marginal returns to income.

9. I’m all for going beyond the original scope of a post. Honestly, this conversation has convinced me that I should re-do and clarify my argument, and also expand it to include a fair number of the things we’ve discussed.

I believe that, because you are a sociologist, I must make some kind of ribald sociology joke or else they’ll kick me out of the economists’ club…

Let me recap what you are saying. I think I can see where the difference in opinion lies, and this might help to keep us on the same page so that we aren’t talking past each other (which, sadly, happens too often in the intellectual world).

Point 1: People have desires, needs, wants, and so on. This is what I have called demand (although “demand” as a technical term means that a person also has the means to acquire such a good).
Point 2: Interactions with other individuals may cause our subject to change his mind, be it out of envy, expected norms of social interaction, or some other cause.
Point 3: People tend to concern themselves with social behavior after taking care of “basic wants/needs,” however an individual may wish to define this for himself.
Point 4: Social considerations—or perhaps any other kind [this is my addition]—may cause an individual to keep purchasing bigger and better things.
Point 5: In particular, social considerations override a person’s “real” preferences.
Point 6: By giving in to these social considerations and “keeping up with the Joneses,” individuals purchase goods which are actually of a lower value to them than they otherwise would.
Point 7: As a result, when they do get around to purchasing the higher-valued goods, there is an increased level of value at the margin. Therefore, individuals do not behave “rationally.”

If I am missing something in your logic chain, I apologize in advance, but this is what I got. I think that we are both in agreement with points 1-3. I wouldn’t doubt for a moment that people try to “keep up with the Joneses,” and it certainly can influence the choices of individuals. Even with point 4, I don’t have a beef. I think it’s where we get to point 5 that I start quibbling.

My big problem with point 5 is that I think you’re introducing an objective value into individual preference-making decisions. If I am not distorting your points, I believe that you are saying, at least to an extent, that there are relatively more legitimate and less legitimate sources of preference creation, and demand caused by envy or notions of respectability in a particular clique is less legitimate than some other source.

I know that there is some level of normative disgust at such things—after all, “keeping up with the Joneses” is a derogatory phrase—but for the purposes of positive analysis, it’s important to realize that there _is_ a value to such things, and that value is precisely the weight which an individual places on such things. I, honestly not caring what my neighbors have, do not gain much value from emulating them. On the other hand, people who do such things end up doing it because they do have some intrinsic satisfaction from copying social trends. [Note that this can also apply to a lot of other fields, such as advertising and fashion.] If you asked me what I personally think about such things, I’d say it’s absurd and ridiculous, but that’s because I’m trying to apply my personal weighting system on the decisions of others—in other words, I am trying to create an objective measure of which preferences people ought to have, when preferences are actually subjective.

The above is a long way of saying that, although buying a new car just because your neighbors bought one may strike you as an action of little value, it does mean more to the person actually performing the action. Utility, as an idea, is extremely broad—in fact, it’s so broad that I don’t think it’s really all that falsifiable. But because it is broad, it includes all of the sources of intrinsic enjoyment, including those derived from social interaction.

In other words and using a cardinal utility function as a case of extreme simplicity and not out of any argument that such use is good outside of an easier-to-understand example, let’s say that a person has \$20,000 to spend. This would derive 5000 utils over time from saving and investing this money for his retirement. On the other hand, in a complete vacuum, he would gain 3000 utils from purchasing a new car. To continue the example, he has another \$10,000 and can derive 2000 utils from upgrading his house or 1500 from investing the \$10K. We will assume that this is the “last” purchase, in that if the person were to lose \$10,000 somehow, this is the thing that the person would forego before anything else.

So I think that what you’re saying is that a person who sees his neighbor buy a new car will experience increasing marginal returns: 15/100 utils/\$ for the car, but then 20/100 utils/\$ for the house repairs. And if the car versus investment were my only two opportunities, it would be folly to purchase the car as opposed to investing the money, and thus “keeping up with the Joneses” is a wasteful phenomenon which can interfere with analysis, incidentally including my argument above.

What is missing from this, though, is the value of this social interaction. Choosing the car _would_ make sense in the event that social oneupsmanship gives a value of more than 2000 utils. In that case, even though the car itself is only worth 3000 utils, the combined satisfaction comes up to a value of more than 5000, making it a better choice than the investment option. After that, the subsequent house upgrade no longer violates the idea of diminishing marginal returns and all is well in Economicsville once more.

Now, why would the value of social interaction be worth more than 2000 utils? That’s where the economist punts and says, “Hey, it’s their choice; all I’m doing is interpreting it.”

Incidentally, I have not missed the irony that I am using a cardinal utility function in a post in which I criticize cardinal utility functions even when they hide behind a veneer of ordinality…

10. Michael A. Lewis says: